570 



THE BELL SYSTEM TECHNICAL JOURNAL, MAY 1952 



currents (17) of 4.58 may l)e regarded as elements of the subspace (10) 

 of 4.55. We have called this subspace Km • The voltages (14), (15), with 

 vo chosen to make (15) an n-tuple of the form (18) (4.58), are elements of 

 a subspace Vji of V . 



It is evident at once that the scalar product between a current (n + m)- 

 tuple (17) and the (n + m)-tuple (14), (15) (vo properly chosen!) is 

 exactly the same as the scalar product between the current (2n)-tuple 

 (10) and the (27i)-tuple formed from the (n + w)-tuple (14), (15) by 

 adjoining (n — m) zeros to expand (15) to an n-tuple of the form (18). 



n 



di 



n 



[II 



n 



ai 



o M- 



Mad 



Fig. 7 = Construction of Mad from Mad*- The solid terminals are those of 

 Mad*, the open circles those of Mad • 



Now we know that we may regard (15) as an n-tuple of the form (18) 

 by a suitable choice of Vo . But we calculated in 4.57 the scalar product 

 between an arbitrary (2n)-tuple and (14), (15) with an arbitrary vq . The 

 answer was (16) of 4.57. By proper choice of Vo , then, (16) represents the 

 bilinear form (1) above for il/(p). Since (16) is independent* of Vo , it 

 contains in itself the whole of the properties of M{p). 



4.61 To show that M{p) is PR, we need show only that M(p) is sym- 

 metric and that is quadratic form (j, = hi and k = ^in (16)) is a PR 

 function of p (2.09). . 



By their definitions, X(p), L(p), and G~ (p) are all symmetric. Hence 

 if all currents are real, the value of (16) is unchanged by interchanging 

 ji with hi , i = 1, 2, and k with /. Therefore M{p) is symmetric. 



4.62 Henceforth we consider the quadratic from 



* This is the gist of P3 of (I, 7.4). Use of the results of I here would have given 

 a more direct but much less constructive representation of Mad • 



